What is the longest slope $\gamma$ in the complement of Dehn surgery space of a cusped hyperbolic 3-manifold $M$? Here Dehn surgery space is the space of fillings such that the hyperbolic structure on the filling $M(\gamma)$ can be realized as a deformation of the original $M$. 

This question is related to Ken Baker’s question:

[Hyperbolic exceptional fillings of cusped hyperbolic 3-manifolds][1]

However, Ken’s question is interested in the total number of slopes in this complement. This question is focusing on the longest such slope, where length is measured by displacement in boundary of a horoball packing (as in the set up for the $6$-theorem or $2\pi$-Theorem). 

Of course, it is possible that this question as stated does not have a realizable answer, because there is no longest slope. 

Here is a more carefully stated version: 

**What is the largest $L$ such that there exist a family of slopes $\gamma_i$ in manifolds $M_i$ such that $$\lim_{i \to \infty} length(\gamma_i) = L$$ 
and each $M_i(\gamma_i)$ is a hyperbolic manifold such that the hyperbolic structure cannot be realized as a deformation of the hyperbolic structure of $M_i$?** 

Of course, the fact that the 6-theorem is sharp implies that $L\geq 6$. Also the $2\pi$ Theorem says $L\leq 2\pi$. 


  [1]: https://mathoverflow.net/questions/121774/hyperbolic-exceptional-fillings-of-cusped-hyperbolic-3-manifolds